Polynomial Identities Separating CD4 from CD5: Minimal Degree Eight, the Complete Octic Relative Cocharacter, and the Nonic Sign Cascade

Abstract

This preprint establishes pure-multiplication polynomial identities separating the standard Cayley-Dickson inclusion CD4 from CD5 and proves that degree eight is minimal, with an explicit separator and nonzero CD5 witness. The exact octic relative cocharacter has dimension 800: six hook blocks contribute 408 dimensions and three near-hook blocks contribute 392, with no conjugation or sign-twist symmetry. This module requires exactly two S8-orbit generators and equals the primitive octic quotient for CD4, while the primitive octic quotient for CD5 is zero. Cyclic re-rooting resolves the two-dimensional octic sign plane into form and hook lines. In degree nine, the certified relative sign channel has dimension 23, its one-step consequence subspace has dimension 7, and its primitive quotient has dimension 16. The accompanying reproducibility release is a single self-contained repository holding Python and Lean software, compact inputs, and exact certificates, so every object a theorem depends on can be checked offline. Generative AI tools assisted with drafting and revising prose, exploring proof and implementation strategies, source-code review, and verification-workflow support. No model output is treated as mathematical evidence; the author reviewed and validated all claims, proofs, computations, references, code, and final wording and takes full responsibility. AI tools are not authors or contributors.